Calculate the standard deviation for the following data: Class 0-30 30-60 60-90 90-120 120-150 10-180 180-210 Frequency: 9 17 43 82 81 44 24

Explanation

To calculate the standard deviation for the given data, we will follow these steps:

### Step 1: Identify the Classes and Frequencies
We have the following classes and their corresponding frequencies:

| Class | Frequency (f) |
|-----------|---------------|
| 0 - 30 | 9 |
| 30 - 60 | 17 |
| 60 - 90 | 43 |
| 90 - 120 | 82 |
| 120 - 150| 81 |
| 150 - 180| 44 |
| 180 - 210| 24 |

### Step 2: Calculate the Midpoint (x) for Each Class
The midpoint for each class can be calculated as follows:
- For class 0-30, midpoint \( x_1 = \frac{0 + 30}{2} = 15 \)
- For class 30-60, midpoint \( x_2 = \frac{30 + 60}{2} = 45 \)
- For class 60-90, midpoint \( x_3 = \frac{60 + 90}{2} = 75 \)
- For class 90-120, midpoint \( x_4 = \frac{90 + 120}{2} = 105 \)
- For class 120-150, midpoint \( x_5 = \frac{120 + 150}{2} = 135 \)
- For class 150-180, midpoint \( x_6 = \frac{150 + 180}{2} = 165 \)
- For class 180-210, midpoint \( x_7 = \frac{180 + 210}{2} = 195 \)

### Step 3: Calculate \( f \cdot x \) and \( f \cdot x^2 \)
Now, we calculate \( f \cdot x \) and \( f \cdot x^2 \) for each class.

| Class | Frequency (f) | Midpoint (x) | \( f \cdot x \) | \( f \cdot x^2 \) |
|-----------|---------------|--------------|------------------|--------------------|
| 0 - 30 | 9 | 15 | \( 9 \cdot 15 = 135 \) | \( 9 \cdot 15^2 = 2025 \) |
| 30 - 60 | 17 | 45 | \( 17 \cdot 45 = 765 \) | \( 17 \cdot 45^2 = 7650 \) |
| 60 - 90 | 43 | 75 | \( 43 \cdot 75 = 3225 \) | \( 43 \cdot 75^2 = 16125 \) |
| 90 - 120 | 82 | 105 | \( 82 \cdot 105 = 8610 \) | \( 82 \cdot 105^2 = 86100 \) |
| 120 - 150| 81 | 135 | \( 81 \cdot 135 = 10935 \) | \( 81 \cdot 135^2 = 1093500 \) |
| 150 - 180| 44 | 165 | \( 44 \cdot 165 = 7260 \) | \( 44 \cdot 165^2 = 1197900 \) |
| 180 - 210| 24 | 195 | \( 24 \cdot 195 = 4680 \) | \( 24 \cdot 195^2 = 912600 \) |

### Step 4: Calculate Summations
Now we sum up the values of \( f \), \( f \cdot x \), and \( f \cdot x^2 \):

- \( \Sigma f = 9 + 17 + 43 + 82 + 81 + 44 + 24 = 300 \)
- \( \Sigma (f \cdot x) = 135 + 765 + 3225 + 8610 + 10935 + 7260 + 4680 = 35610 \)
- \( \Sigma (f \cdot x^2) = 2025 + 7650 + 16125 + 86100 + 1093500 + 1197900 + 912600 = 4769100 \)

### Step 5: Calculate the Variance
Using the formula for variance:

\[ \sigma^2 = \frac{\Sigma (f \cdot x^2)}{N} - \left( \frac{\Sigma (f \cdot x)}{N} \right)^2 \]

Where \( N = \Sigma f = 300 \):

\[ \sigma^2 = \frac{4769100}{300} - \left( \frac{35610}{300} \right)^2 \]

Calculating each part:

1. \( \frac{4769100}{300} = 15897 \)
2. \( \frac{35610}{300} = 118.70 \)
3. \( (118.70)^2 = 14092.49 \)

Now substituting back into the variance formula:

\[ \sigma^2 = 15897 - 14092.49 = 804.51 \]

### Step 6: Calculate the Standard Deviation
Finally, take the square root of the variance to find the standard deviation:

\[ \sigma = \sqrt{804.51} \approx 28.37 \]

### Final Answer
The standard deviation of the given data is approximately **28.37**.